The sheaves of p-adic nearby cycles on varieties over mixed characteristic local rings can be described (in some stable range and up to universal constants) by syntomic sheaves, i.e., by relative version of the Fontaine functor applied to crystalline cohomology. Over an algebraically closed field or for large p this is a classical result due to Bloch-Kato, Kato, Kurihara, Tsuji.

I will describe an approach to this comparison result that uses (\phi,\Gamma)-modules as a bridge between the etale cohomology and syntomic cohomology. This allows to extend the above result to all primes p. A global comparison theorem for semistable proper (formal) schemes follows. This is a joint work with Pierre Colmez.